Density

What this lesson covers in the specification

Incandio is aligned to this specification. It is not published by, endorsed by or affiliated with Pearson, and it reproduces none of Pearson's wording — the statement numbers are given so you can check every lesson against your own copy.

• 5.1 — Use the units °C, K, J, kg, kg/m³, m, m², m³, m/s, m/s², N and Pa
• 5.3 — Use density = mass ÷ volume
• 5.4 — Practical: investigate density using direct measurements of mass and volume (required practical)

1 · Understand it

No exam language yet. The only question this section answers is: do I actually understand what is happening?

Ask whether lead is heavier than feathers and the question is badly put, because a lorry-load of feathers outweighs a marble of lead. What people mean is that lead has more mass packed into each unit of volume — and that is DENSITY: the mass of a material divided by the volume it occupies.

The equation is density = mass ÷ volume, and it is worth reading the unit as an instruction. Kilograms per cubic metre, kg/m³, means exactly what it says: how many kilograms there would be in one cubic metre of the stuff. Water is about 1000 kg/m³, so a cubic metre of water is a tonne. Lead is about 11 300 kg/m³, and air at room temperature is about 1.2 kg/m³.

Density is a property of the MATERIAL, not of the object. Cut a copper block in half and each half has half the mass and half the volume, so the density is unchanged — which is why it is useful for identifying what something is made of, and why a question can give you a density and expect you to work out what the material is.

Think of it like people in a room rather than people in a building

Asking how crowded a building is by counting the people in it tells you very little — a thousand people in a stadium is empty, and a thousand in a house is a catastrophe. What you want is people per room, or per square metre. Density is that: not the total, but the total divided by the space it is in. This also explains the two ways a material can be dense. Either its particles are individually heavy, as with lead, or they are packed very tightly together — and in a gas, where the particles are far apart, the density is tiny even though the particles themselves are perfectly ordinary.

Density calculated, and used to identify a material

(a) A block has a mass of 480 g and measures 4.0 cm by 5.0 cm by 3.0 cm. Find its density in kg/m³. (b) A 2.0 kg sample of the same material — what volume does it occupy?

1. (a) Volume = 4.0 × 5.0 × 3.0 = 60 cm³. Convert: 60 cm³ = 60 ÷ 1 000 000 = 0.000060 m³.
2. Convert the mass: 480 g = 0.480 kg.
3. density = mass ÷ volume = 0.480 ÷ 0.000060 = 8000 kg/m³.
4. (b) volume = mass ÷ density = 2.0 ÷ 8000 = 0.00025 m³, which is 250 cm³.
5. 8000 kg/m³ is close to the density of steel, so the block is probably steel rather than aluminium at 2700 or lead at 11 300.

Answer: 8000 kg/m³, and 0.00025 m³ for a 2.0 kg sample.

The conversion in that worked example is the single biggest source of lost marks in this topic, and it is worth doing slowly once. There are 100 cm in a metre, so a cubic metre is 100 × 100 × 100 = 1 000 000 cubic centimetres. To go from cm³ to m³ you therefore divide by a MILLION, not by a hundred. A density in g/cm³ can also be converted straight to kg/m³ by multiplying by 1000, which is why water is 1.0 g/cm³ and 1000 kg/m³ at the same time.

Measuring the volume — three cases, three methods

1. A REGULAR SOLID — a cube, a cuboid or a cylinder. Measure the dimensions with a ruler or vernier callipers and calculate the volume from them.
2. An IRREGULAR SOLID that sinks. Use displacement: lower it into a measuring cylinder of water on a thread and record the rise in level, or use a displacement can and collect the overflow. The volume of water displaced equals the volume of the object.
3. A LIQUID. Measure its mass by weighing an empty measuring cylinder, then the cylinder with the liquid in it, and subtracting. Read the volume directly from the cylinder at eye level.
4. In every case the mass is found with a balance, and the same equation does the rest.

The displacement method is worth a moment, because it is the oldest idea in this topic. An object lowered into water pushes aside exactly its own volume of water, whatever shape it is — which converts an impossible measurement into an easy one. It is the principle behind the story of Archimedes and the crown, and although the story itself is probably a later embellishment, the physics in it is entirely sound.

Finally, statement 5.1 asks for the units of this whole topic, and it is worth collecting them: DEGREES CELSIUS and KELVIN for temperature, JOULES for energy, KILOGRAMS for mass, KILOGRAMS PER CUBIC METRE for density, METRES for length, SQUARE METRES for area, CUBIC METRES for volume, METRES PER SECOND for speed, METRES PER SECOND SQUARED for acceleration, NEWTONS for force, and PASCALS for pressure. A pascal is a newton per square metre, which is the next page in one phrase.

2 · Grade 9 Notes

A different job from the section above. You have already understood it; this is the precise set of things to LEARN — definitions to reproduce word for word, processes in order, equations with units, and the answers that score full marks.

density = mass ÷ volume

density in kilograms per cubic metre (kg/m³), mass in kilograms (kg), volume in cubic metres (m³)

Units: kg/m³, kg, m³. Rearranged: mass = density × volume and volume = mass ÷ density. 1 g/cm³ = 1000 kg/m³.

Learn this definition · Density

The mass of a material per unit volume. It is a property of the material rather than of the object, so cutting an object in half does not change it.

Statement 5.1 — the units of this topic

kg/m³

Density. Water is 1000 kg/m³, air about 1.2 kg/m³, lead about 11 300 kg/m³.

Pa (pascal)

Pressure. One pascal is one newton per square metre.

m², m³

Area and volume. A cubic metre is 1 000 000 cubic centimetres.

°C and K

Temperature in degrees Celsius and in kelvin; 0 K is −273 °C.

J, kg, N

Energy, mass and force.

Required practical 5.4 — measuring density from mass and volume

1. REGULAR SOLID: measure the length, width and height with a ruler or vernier callipers and calculate the volume.
2. Find the mass on a balance, zeroed before use, and calculate density = mass ÷ volume.
3. IRREGULAR SOLID: find its mass on the balance first, while it is dry.
4. Part-fill a measuring cylinder with water, record the level, lower the object in on a thread until fully submerged, and record the new level.
5. The volume of the object is the difference between the two readings; calculate the density as before.
6. LIQUID: weigh an empty measuring cylinder, add the liquid, weigh again and subtract to find the mass of the liquid, reading its volume from the scale at eye level.

Variables

Independent (changed) — The material or object whose density is being found
Dependent (measured) — The density calculated, in kg/m³

Sources of error

Getting the units right — the conversion that costs most marks

1. 1 m = 100 cm, so 1 m³ = 100 × 100 × 100 = 1 000 000 cm³.
2. To convert cm³ to m³, DIVIDE BY 1 000 000.
3. To convert g to kg, divide by 1000.
4. To convert a density in g/cm³ to kg/m³, MULTIPLY BY 1000.
5. Check: water is 1.0 g/cm³ and 1000 kg/m³, which is the same thing said twice.

Measuring volume — which method for which object

• REGULAR SOLID — measure the dimensions and calculate
• IRREGULAR SOLID that sinks — displacement: the rise in water level, or the overflow from a displacement can
• LIQUID — read directly from a measuring cylinder at eye level, and find its mass by weighing before and after
• The object must be FULLY SUBMERGED, and bubbles must be tapped off, or the volume comes out too large

Model answer [4 marks]

A metal cube of side 2.0 cm has a mass of 21.6 g. Calculate its density in kg/m³. [4]

The volume is 2.0 × 2.0 × 2.0 = 8.0 cm³, which is 8.0 ÷ 1 000 000 = 0.0000080 m³. The mass is 21.6 g = 0.0216 kg. Density = mass ÷ volume = 0.0216 ÷ 0.0000080 = 2700 kg/m³, which is the density of aluminium.

Model answer [4 marks]

Describe how you would find the density of a small irregular stone. [4]

Find the mass of the dry stone using a balance that has been zeroed. Part-fill a measuring cylinder with water and record the level at eye level. Lower the stone into the cylinder on a thread until it is completely submerged, tapping it to release any air bubbles, and record the new water level. The difference between the two readings is the volume of the stone, because it displaces its own volume of water. Calculate the density by dividing the mass by the volume, converting both into kilograms and cubic metres.

Not this: A large object has a greater density than a small one made of the same material.

This: Density is a property of the MATERIAL. A large object has more mass AND more volume in the same proportion, so the density is identical — which is exactly why density can be used to identify what something is made of.

Mark-losing trap. cm³ to m³ means dividing by A MILLION, not by 100. This is the commonest error in the topic.

Mark-losing trap. Density is a property of the MATERIAL — cutting an object up does not change it.

Mark-losing trap. The object must be FULLY submerged for displacement, and bubbles tapped off.

Mark-losing trap. 1 g/cm³ = 1000 kg/m³. Water is both, which is a useful check on any conversion.

3 · Prove it — the five questions

The five questions climb Grade 6 → Grade 7 → Grade 8 → Grade 9 → Grade 9 challenge, and are marked inside Incandio on your own device, by rule, with an authored diagnosis of the mistake you actually made. The mark schemes stay in the app so that the practice is worth doing; the questions themselves are here.

1. Grade 6 · Calculate [2 marks] — A block has a mass of 60 kg and a volume of 0.030 m³. Calculate its density.
2. Grade 7 · State [2 marks] — A solid copper cube is cut exactly in half. What happens to the density of each half?
3. Grade 8 · Calculate [4 marks] — A rectangular block measures 5.0 cm by 4.0 cm by 2.0 cm and has a mass of 216 g. Calculate its density in kg/m³.
4. Grade 9 · Describe [6 marks] — Select every statement that belongs in a full-mark description of how to find the density of a small irregular stone.
5. 9+ · Analyse [6 marks] — A student measures a stone's density and gets 4200 kg/m³, but the accepted value for that rock is 2600 kg/m³. They conclude the stone must be a different mineral. Select every statement that belongs in a full-mark analysis.

The people behind this science

Two ways into the same idea — the one who turned an impossible measurement into an easy one, and the one who built a better instrument for the same problem. Inside Incandio each of them answers knowing exactly which lesson you have just finished.

Archimedes — the one who turned an impossible measurement into an easy one

The displacement method on this page is the oldest idea in the topic and it is attributed to him. The problem set by King Hiero was whether a crown was pure gold, which requires its density, which requires its volume — and a crown has no dimensions you can measure with a rule. Submerging it converts the impossible into the trivial, because any object pushes aside exactly its own volume of water. He is also the right person to ask about the famous story, because the version everyone knows comes from Vitruvius two centuries later and the method it describes would have been very hard to perform accurately.

• “How would you actually find the volume of a crown?”
• “What was King Hiero really asking you to find out?”
• “Is the story about the bath true?”
• “Why does an object push aside exactly its own volume?”
• “What could you tell about a material from its density alone?”

Galileo Galilei — the one who built a better instrument for the same problem

Galileo's first published work, La Bilancetta of 1586, was written because he thought the bath story was nonsense — the change in water level for a crown would be far too small to measure reliably. He designed a hydrostatic balance instead: weigh the object in air, then weigh it again while it hangs submerged, and the difference gives the upthrust and hence the volume, far more precisely than any water level could. It is a young man's paper about doing an old measurement properly, and it is exactly the reasoning behind the errors section of this page's practical.

• “Why did you think the bath story could not be right?”
• “How does weighing something underwater tell you its volume?”
• “What is wrong with just measuring the rise in the water?”
• “Why does the precision of a measurement matter so much?”
• “What was it like publishing your first paper as a young man?”

Then defend it

On Incandio a lesson is not finished when the questions come out right. You teach the idea back to Ember, an AI apprentice who asks the awkward question, and then you argue it against Archimedes in a structured debate marked against descriptors you can read before you enter. Learn it, teach it, then defend it — all three happen on this page once the app loads.

Carry on through the course

• Previous lesson: Generating Electricity
• Next lesson: Pressure, and Pressure in Liquids
• Pressure, and Pressure in Liquids — Why a drawing pin works, why pressure at a point pushes every way at once, and why depth is the only thing that matters underwater.
• Particles and Changes of State — What the particles are doing in each state — and why a boiling pan stays at 100 °C however hard you heat it.
• Specific Heat Capacity — Why the sand burns your feet and the sea is freezing on the same afternoon.
• Gas Pressure and Absolute Zero — Where the pressure of a gas actually comes from, and why the temperature scale has to start somewhere else.
• The Kinetic Theory of Gases — One picture — fast random molecules bouncing off walls — that predicts everything a gas does.
• All of Physics · Incandio Science